To subtract these radicals together their radicands must be the same:

7\( \sqrt{45} \) - 4\( \sqrt{5} \)
7\( \sqrt{9 \times 5} \) - 4\( \sqrt{5} \)
7\( \sqrt{3^2 \times 5} \) - 4\( \sqrt{5} \)
(7)(3)\( \sqrt{5} \) - 4\( \sqrt{5} \)
21\( \sqrt{5} \) - 4\( \sqrt{5} \)

Now that the radicands are identical, you can subtract them:

A prime number is an integer greater than 1 that has no factors other than 1 and itself. Examples of prime numbers include 2, 3, 5, 7, and 11.

To multiply terms with radicals, multiply the coefficients and radicands separately:

7\( \sqrt{2} \) x 9\( \sqrt{7} \)
(7 x 9)\( \sqrt{2 \times 7} \)
63\( \sqrt{14} \)

To convert a negative exponent to a positive exponent, calculate the positive exponent then take the reciprocal.

To subtract these fractions, first find the lowest common multiple of their denominators. The first few multiples of 3 are [3, 6, 9, 12, 15, 18, 21, 24, 27, 30] and the first few multiples of 11 are [11, 22, 33, 44, 55, 66, 77, 88, 99]. The first few multiples they share are [33, 66, 99] making 33 the smallest multiple 3 and 11 share.

Next, convert the fractions so each denominator equals the lowest common multiple:

\( \frac{4 x 11}{3 x 11} \) - \( \frac{3 x 3}{11 x 3} \)

\( \frac{44}{33} \) - \( \frac{9}{33} \)

Now, because the fractions share a common denominator, you can subtract them:

\( \frac{44 - 9}{33} \) = \( \frac{35}{33} \) = 1\(\frac{2}{33}\)